Ratio Estimators Using Coefficient of Variation and Coefficient of Correlation

This paper introduces ratio estimators of the population mean using the coefficient of variation of study variable and auxiliary variables together with the coefficient of correlation between the study and auxiliary variables under simple random sampling and stratified random sampling. These ratio estimators are almost unbiased. The mean square errors of the estimators and their estimators are given. Sample size estimation in both sampling designs are presented. An optimal sample size allocation in stratified random sampling is also suggested. Based on theoretical study, it can be shown that these ratio estimators have smaller MSE than the unbiased estimators. Moreover, the empirical study indicates that these ratio estimators have smallest MSE compared to the existing ones.


Introduction
Consider a population of N units with observations   i i x , y for   i 1, 2, , N where i y is a value of study variable and i x is a value of auxiliary variable.Under simple random sampling without replacement, an unbiased estimator of the population mean is the sample mean . The variance of the unbiased estimator is where n f N  and where X and x are the population and sample means of the auxiliary variable, respectively.The efficiency of the ratio estimator depends on the coefficient of variation of and coefficient of variation of study variable   y C .Murthy (1964) has suggested that if   x y C 2C , the ratio estimator performs better than the unbiased estimator where  is the correlation coefficient between x and y .The approximate bias and mean square error (MSE) of the ratio estimator are as follows: . When the Sissodia and Dwivedi (1981) has proposed a modified ratio estimator, . Sampath (2005) used the coefficient of variation of the study variable to improve the unbiased estimator as . The approximate bias and MSE of this estimator are   (1.7) In addition, there are several authors, such as Upadhyaya and Singh (1999), Singh and Tailor (2003), who have developed various ratio estimators under simple random sampling.
n is a sample size in stratum h and S is the variance of the study variable in stratum h.There are two types of ratio estimators in stratified random sampling, namely combined and separate ratio estimators.
The combined ratio estimator is given by , where x is an unbiased estimator of the population mean X and h x is the sample mean of auxiliary variable in stratum h (Cochran, 1977).The approximate mean squared error of the combined ratio estimator is is the population ratio, (1.10) The separate ratio estimator is given by . The approximate MSE of the separate ratio estimator can be given by is the population ratio in stratum h.The approximate bias of the separate ratio estimator is (1.12) Kadilar and Cingi (2003) have proposed several combined ratio estimators.The simplest one based on the Sissodia and Dwivedi (1981) estimator is defined as is the coefficient of variation of the auxiliary variable in stratum h.The MSE and bias of this estimator are approximated as follows: where . Kadilar and Cingi (2005) have improved the combined ratio estimator in stratified random sampling based on the estimator introduced by Prasad (1989).However, these estimators depend on several unknown parameters and therefore aredifficult to use.
In Sections 2 and 3, the ratio estimators based on the coefficient of variation and correlation in simple random sampling and stratified random sampling are introduced, respectively.The approximate bias and MSE of the estimator are derived.An estimator of the MSE is given.The sample size estimation and an optimal allocation of sample size in stratified random sampling is presented.The comparison of the efficiency between the proposed estimator and unbiased estimator is theoretically provided.Hypothetical populations are used to compare the properties of the presented estimators with the existing ones.

Parameter Estimation
Consider the following ratio estimator for the population mean of the study variable, where c is a real constant to be determined such that the   c MSE y is minimized.Note that when c is equal to 0, this estimator is reduced to the usual ratio estimator and when c is equal to x C this estimator become the estimator of Sissodia and Dwivedi (1981).To obtain the MSE and bias of the estimator (2.1), let . The estimator c y can be written as . Using Taylor series approximation, we obtain y Y 1 e e e e e ... .When the terms of degree greater than two are ignored, we get the approximate bias of the estimator c y as . Similarly, Taylor's formula can be used to approximate MSE of the estimator as for c in (2.1), (2.2) and (2.3) and using algebra, we obtain the optimum estimator, its bias and MSE as follows, Note that the optimum estimator is almost unbiased and its MSE is always smaller than the variance of the unbiased estimator.In addition, the optimum estimator can be applied for both populations with positive and negative coefficient of correlation.For a sample estimate of the MSE, one can substitute the sample estimate of where 2 y s is the sample variance of the study variable.

Sample Size Estimation
Sample size estimation is one of the important aspects in sample surveys.If the sample size is too small, the sampling error may be too large.However, too large sample size implies a waste of resources.We would like to specify a sample size that is sufficiently large to ensure a high probability that the estimate closes to the where . If the population size N is large relative to the sample size n, the formula of the sample size reduces to 0 n .

Comparison of Efficiency
In this section, the properties of the estimators in simple random sampling are compared.The relative efficiency of the optimum estimator and unbiased estimator is considered as follows: (2.9) This shows that the optimum estimator is always more efficient than the unbiased estimator because The efficiency depends on the coefficient of correlation:if the coefficient of correlation increases then the efficiency also increases.
To compare the properties of the optimum estimator with the others, we consider hypothetical populations with vary characteristics.In this work, the coefficients of correlation in the populations are    0.1, 0.2, , 0.9 .In each population, the coefficients of variations are 0 2 . and the population means 5 000  X , , 5 000  Y , .With varying sample sizes, the biases and MSEs of the estimators are given in Table 1 and Table 2, respectively.The biases and MSEs are computed by the formulas in the previous sections.In Table 1, as expected, the absolute bias of the unbiased and optimum ratio estimators are always equal to 0. The estimator S y has the largest absolute bias among the compared estimators.The bias of the estimator S y is negative because the estimator is constructed by using a constant in which its value less than 1 multiplying the unbiased estimator.The bias of the estimator S y does not depend the coefficient of correlation.Observe that the absolute biases of the estimator SD y are smaller than of the estimator R y .Given a sample size, when the coefficient of correlation increases the absolute bias of the two estimators R y and SD y decrease.Given a coefficient of correlation, the absolute bias of S y , R y and SD y decrease when the sample size increases.Table 2 shows that the optimum ratio estimator has smallest MSE among the compared estimators.The MSEs of the two estimators y and S y do not depend on the coefficient of correlation.When 0 5   .the MSEs of the two estimators R y and SD y are less than those of the unbiased estimator.Given a sample size, when the coefficient of correlation increases the MSEs of the three estimators R y , SD y and c* y decrease.Given a coefficient of correlation, the MSEs of all estimators decrease when the sample size increases.
Since this estimator is constructed from the optimum ratio estimator, we call this estimator "optimum separate ratio estimator".To obtain the MSE and bias of the optimum separate ratio estimator, applying the MSE and bias of under simple random sampling to draw in stratum h,yields MSE y , we substitute the sample estimates to obtain Note that the bias of the optimum separate ratio estimator is the cumulative bias of an optimum ratio estimate in each stratum which closes to zero.In addition, we found that the MSE of the estimator is also smaller than the variance of the unbiased estimator in (1.8).

Optimum Sample Size Allocation
Given a total sample size n and using the optimum separate ratio estimator, one may choose how to allocate the sample size among the L strata.In this section, the allocation scheme which minimizes the MSE of the estimator by fixing the total sample size is considered.That is, we need the values of 1 2  L n , n , , n which minimize (3.5) Thus, the optimum scheme allocates larger sample sizes to strata with larger variances and larger stratum sizes but smaller sample sizes to strata with larger coefficients of correlation.

Sample Size Estimation
The formula (3.5) gives h n in terms of n , but in practice, we do not yet know what value of n is.This section presents a formula for the determination of n under the optimum sample size allocation.It is assumed that the optimum separate ratio estimate has a specified mean squared error M .When Solving for n, we have

Comparison of Efficiency
In this section, we compare the properties of the proposed optimum separate ratio estimator with the existing ones in stratified random sampling.The relative efficiency of the optimum separate ratio estimator and unbiased estimator is This shows that the optimum separate ratio estimator is always more efficient than the unbiased estimator.The efficiency depends on the coefficient of correlation in stratum.If the correlation coefficient increases then the efficiency also increases.
To compare the properties of the optimum separate ratio estimator with the other estimators, we consider the following hypothetical populations.Each population consists of N = 50,000 units and is divided into L = 2 strata of which sizes are N and optimal allocation as in (3.5).The biases and MSEs of the estimators are given in Table 3 and Table 4, respectively.The biases and MSEs are computed by the formulas in the above sections.
In  is the coefficient of correlation in the whole population.

Discussion
In simple random sampling, the optimum ratio estimator and its variance estimate depend on the coefficient of variation, the coefficient of correlation and the mean of the auxiliary variable in the whole population.Similarly, the optimum separate ratio estimator and its variance estimate are in terms of the coefficients of variation, the coefficient of correlation and the means of the auxiliary variable in all strata.In practice, sample estimates of these parameters may be used to substitute in the formulas of these estimates.
In simple random sampling, the relative efficiency of the optimum ratio estimator and the unbiased estimator depends on the coefficient of correlation  .When the coefficient of correlation between the study and auxillary variable is weak, then the relative efficiency will be low so that the unbiased estimator is almost as good as the ratio estimators.For example, if 0 2   ., then   1 042  c* e y, y .
. This means that if we increase the sample size about 4.2%, the unbiased estimator will have the most efficiency among the estimators in the class of . Therefore, in case of 0 2   .we suggest using the unbiased estimator because it uses only the information of the study variable and we do not need to collect the data of the auxillary variables.In stratified random sampling, when . , we also recommend the unbiased estimator.For future studies, we can consider applying the ratio estimator in adaptive sampling schemes as suggested by Thompson (1990) and Sangngam (2013) for.
parameter.Under simple random sampling, the population mean of the study variable   Y is estimated with the optimum estimator c* y .To obtain the desired sample size, one can specify the margin of error d and the probability  such that    .Under some technical conditions as shown in Scott and Wu (1981) and Hajek (1960), we can show that c* y is asymptotically normal distributed with mean Y and variance   c*MSE y.To obtain the absolute precision, we can find a value of n that satisfies large and the technical conditions in Scott and Wu (1981) hold, we can show that the estimator RS _ C y has also the asymptotic normal distribution.If the margin of error d has been given, then

Table 1 .
Biases of the estimators in simple random sampling

Table 2 .
MSEs of the estimators in simple random sampling Cand  xh in stratum h are known, the separate ratio estimator can be modified as

Table 4
Table 3, the absolute biases of the unbiased and optimum separate ratio estimators are always equal to 0. The absolute biasof the estimator RS y is smallest among the compared estimators.The absolute bias of the estimator KCy is smaller than that of the estimator RC y .Given an sample size allocation, when the coefficient of correlation increases the absolute biases of the three estimators RC presents that the optimum separate ratio estimator gives the smallest MSE among the compared estimators.Observe that the MSE of the unbiased estimator st y does not depend on the coefficient of correlation because it does not use the information of the auxillary variable.

Table 3 .
Biases of the estimators in stratified random sampling